I want to have the best person available and I don't mind if that person is male or female
Nice attempt to derail from the facts.
The issue is NOT male or female HCPs.
It is male sexed people who come in place of a female sexed people (that you have requested or is the widely accepted norm. Or in any scenario where you would rightfully expect to be advised that it was a male HCP).
This male then tells you they are a woman, when you and they both know full well the truth. They may or may not wear feminine clothing, or they may be a 6' alpha male brick shithouse in chinos. Appearance is irrelevant to their claim of womanhood.
You are then required to go along with this pretence that they are a woman purely for their benefit, and potentially at harm to you. Raising concern gets you labelled as a troublemaker, appt waster, bigot, transphobe. You may even lose your place in the queue for whatever medical service you are using.
Up till now, that has not been the case, for the obvious and longstanding accepted reasons of privacy, dignity and safety. It's only now that people are saying it should be changed TO this. Why are you so keen on that?